A model of the generic Vopěnka principle in which the ordinals are not Mahlo

Abstract

The generic Vopěnka principle, we prove, is relatively consistent with the ordinals being non-Mahlo. Similarly, the generic Vopěnka scheme is relatively consistent with the ordinals being definably non-Mahlo. Indeed, the generic Vopěnka scheme is relatively consistent with the existence of a ${\Delta }_2$-definable class containing no regular cardinals. In such a model, there can be no ${\Sigma }_2$-reflecting cardinals and hence also no remarkable cardinals. This latter fact answers negatively a question of Bagaria, Gitman and Schindler.

arXiv comment19 pages. Commentary concerning this paper can be made at http://jdh.hamkins.org/generic-vopenka-ord-not-mahlo. v2 has minor revisions

Cited by in corpus (1)

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